Q-DISCRETE PAINLEVÉ EQUATIONS: THEIR HIERARCHIES AND PROPERTIES

Abstract

The main objective of this thesis is to derive hierarchies of q-discrete PainelevA© equations. Some of the important properties of these hierarchies will also be given, namely Lax pairs, BA¤cklund transformations, solutions of their asso- ciated linear problems for special values of parameters and their symmetry groups. To construct these hierarchies, we apply a geometric reduction and a stair- case method on a multi-parameteric generalized lattice modified Korteweg-de Vries equation. In addition, the property of consistency around the cube is used in order to find BA¤cklund transformations. Starting with the base case of q-discrete second, third and fourth PainlevA© equations on A5 initial-values surface, new hierarchies of q-discrete third and fourth PainlevA© equations are discovered, and we also rediscover the hierarchy of q-discrete second PainlevA© equation. In this thesis, we provide the Lax pairs for each member in these hierarchies. Using the consistency around the cube, we also provide the BA¤cklund transformation for the entire hierarchy of q-discrete second and third PainlevA© hierarchies. We generate a hierarchy of special solutions starting with seed solutions for q-discrete second and third PainlevA© hierarchies. An assumption made is that particular parameter values would enable the ability to diagonalize the Lax pair. As a consequence, we found that the as- sociated linear problem for the three hierarchies can be solved in terms of q-Gamma function. Furthermore, the hierarchy of q-discrete fourth PainlevA© hierarchy can be reduced to one equation that can be linearlized to become Riccati equation which has hypergeometric special solutions. Finally, we investigated the affine Weyl group structure of the symmetry group for each hierarchy. In this thesis, we construct the explicit representation of the symmetry group for the first and second member of these hierarchies. The collection of new hierarchies, their Lax pairs, BA¤cklund transforma- tions, the resultant symmetry groups and special solutions comprise the new results of this thesis. This thesis contains material published in [10] in collabo- ration with N. Joshi and D. Tran and myself. The material of this paper is pre- sented in Chapter 3, and is related to qPII and qPIII hierarchies, their Lax pair and examples. In Chapter 4, BA¤cklund transformation of qPII and qPIII hierarchies, includes material from the above-mentioned paper. Similarly, Chapter 5 reports on results about solutions of the linear problem from the above paper. However, we emphasize that all the results about qPIV through out the thesis are completely new and unpublished. Chapter 6 includes unpublished material even for qPII and qPIII hierarchies.